Question

A line passes through the point (3 comma 1 comma negative 8 )​, and is parallel to the vector 5 Bold i plus 3 Bold j plus 3 Bold k. Find the standard parametric equations for the​ line, written using the components of the given vector and the coordinates of the given point.

Answer 1

The line $\ell$ passing through (3, 1, -8) and running parallel to the vector <5, 3, 3> has the vector equation

$\ell=(3\,\vec\imath+\vec\jmath-8\,\vec k)+t(5\,\vec\imath+3\,\vec\jmath+3\,\vec k)$

or

$\ell=(3+5t)\,\vec\imath+(1+3t)\,\vec\jmath+(3t-8)\,\vec k$

and from this we can extract a set of parametric equations for the line:

$\begin{cases}x(t)=5t+3\\y(t)=3t+1\\z(t)=3t-8\end{cases}$

where $t\in\Bbb R$.